Sally Dawson, BNLStandard Model and Higgs PhysicsFNAL LHC School, 2006

Introduction to the Standard Model Review of the SU(2) x U(1) Electroweak theory Experimental status of the EW theory Constraints from Precision Measurements

Searching for the Higgs Boson The Importance of the TeV Scale

Lecture 1

Introduction to the Standard Model Just the SU(2) x U(1) part of it….

Some good references: Chris Quigg, Gauge Theories of the Strong, Weak,

and Electromagnetic Interactions Michael Peskin, An Introduction to Quantum Field

Theory David Rainwater, Sally Dawson, TASI2006,

http://quark.phy.bnl.gov/~dawson/tasi06

Tevatron LHC LHC Upgrade ILC

2006 2007 2012

Collider Physics Timeline

What We Know The photon and gluon appear to be

massless The W and Z gauge bosons are heavy

MW=80.404 0.030 GeV MZ =91.1875 0.0021 GeV

There are 6 quarks Mt=172.5 2.3 GeV (171.4 2.1 GeV, ICHEP 2006) Mt >> all the other quark masses

What We Know There appear to be 3 distinct neutrinos

with small but non-zero masses The pattern of fermions appears to

replicate itself 3 times Why not more?

Abelian Higgs Model Why are the W and Z boson masses non-zero?

U(1) gauge theory with single spin-1 gauge field, A

U(1) local gauge invariance:

Mass term for A would look like:

Mass term violates local gauge invariance We understand why MA = 0

AAF

FFL

4

1

)()()( xxAxA

AAmFFL 2

2

1

4

1

Gauge invariance is guiding principle

Abelian Higgs Model, 2 Add complex scalar field, , with charge – e:

Where

L is invariant under local U(1) transformations:

)(4

1 2

VDFFL

2222)(

V

ieAD

)()(

)()()()( xex

xxAxAxie

AAF

Abelian Higgs Model, 3

Case 1: 2 > 0 QED with MA=0 and m= Unique minimum at =0

)(4

1 2

VDFFL

2222)(

V

ieAD

By convention, > 0

Abelian Higgs Model, 4 Case 2: 2 < 0

Minimum energy state at:

2222)( V

22

2 v

Vacuum breaks U(1) symmetry

Aside: What fixes sign (2)?

Abelian Higgs Model, 5 Rewrite

L becomes:

Theory now has: Photon of mass MA=ev Scalar field h with mass-squared –22 > 0 Massless scalar field (Goldstone Boson)

hve vi

2

1

)nsinteractio,(2

1

22

1

24

1 2222

h

hhhAAve

evAFFL

and h are the 2 degrees of freedom of the complex Higgs field

Abelian Higgs Model, 6 What about mixed -A propagator?

Remove by gauge transformation

field disappears

We say that it has been eaten to give the photon mass field called Goldstone boson This is Abelian Higgs Mechanism This gauge (unitary) contains only physical particles

ev

AA1'

)(2

1

24

1 22

hVhhAAve

FFL

Higgs Mechanism summarized

Spontaneous breaking of a gauge theoryby a non-zero VEV of a scalar field results in the disappearance of a Goldstone boson and its transformation into the longitudinal component of a massive gauge boson

R gauges

)1(2222

AA Mk

kkg

Mk

i

22hMk

i

22AMk

i

Mass of Goldstone boson depends on =1: Feynman gauge with massive =0: Landau gauge: Unitarity gauge

Higgs, h

Goldstone Boson, , or Faddeev-Popov ghost

Gauge Boson, A

Non-Abelian Higgs Mechanism

Vector fields Aa(x) and scalar fields i(x) of

SU(N) group

L is invariant under the non-Abelian symmetry:

a are group generators, a=1…N2-1 for SU(N)

22 )()(

),())((

V

VDDL

jijaa

i i )1(

N

.

.

.1

For SU(2): a=a/2

aa AigD

Non-Abelian Higgs Mechanism, 2 In exact analogy to the Abelian case

a0 0 Massive vector boson + Goldstone boson

a0=0 Massless vector boson + massive scalar field

...)()(...

...)()(...))((

002

2

0

bai

bi

a

bai

bi

a

AAg

AAgDD

Non-Abelian Higgs Mechanism, 3 Consider SU(2) example Suppose gets a VEV:

Gauge boson mass term

Using the property of group generators, a,b=ab/2 Mass term for gauge bosons:

a

a

AigD2

v

0

2

1

baba AA

vvgD

0,0

2

1 22

aamass AA

vgL

8

22

Standard Model Synopsis

Group: SU(3) x SU(2) x U(1)

Gauge bosons: SU(3): G

i, i=1…8 SU(2): W

i, i=1,2,3 U(1): B

Gauge couplings: gs, g, g SU(2) Higgs doublet:

ElectroweakQCD

SM Higgs Mechanism Standard Model includes complex Higgs SU(2)

doublet

With SU(2) x U(1) invariant scalar potential

If 2 < 0, then spontaneous symmetry breaking Minimum of potential at:

Choice of minimum breaks gauge symmetry Why is 2 < 0?

043

21

2

1

i

i

22 )( V

v

0

2

1

More on SM Higgs Mechanism Couple to SU(2) x U(1) gauge bosons

(Wi, i=1,2,3; B)

Gauge boson mass terms from:

BYg

iWg

iD

VDDL

ii

S

22

)()()('

...)()()(8

...

...0

))((,08

1...)(

232222122

BggWWgWgv

vBggWBggWvDD bbaa

Justify later: Y=1

More on SM Higgs Mechanism

With massive gauge bosons:

W = (W

1 W2) /2

Z 0 = (g W3 - g'B )/ (g2+g'2)

Orthogonal combination to Z is massless photon

A 0 = (g' W3+gB )/ (g2+g'2)

MW=gv/2MZ=(g2+g'2)v/2

More on SM Higgs Mechanism, 2

Weak mixing angle defined

Z = - sin WB + cosWW3

A = cos WB + sinWW3

MW=MZ cos W

2222sincos

gg

g

gg

gWW

More on SM Higgs Mechanism

Generate mass for W,Z using Higgs mechanism Higgs VEV breaks SU(2) x U(1)U(1)em

Single Higgs doublet is minimal case Just like Abelian Higgs model

Goldstone Bosons Before spontaneous symmetry breaking:

Massless Wi, B, Complex After spontaneous symmetry breaking:

Massive W,Z; massless ; physical Higgs boson h

z,

Fermi Model Current-current interaction of 4 fermions

Consider just leptonic current

Only left-handed fermions feel charged current weak interactions (maximal P violation)

This induces muon decay

JJGL FFERMI 22

hceJ elept

2

1

2

1 55

e

e GF=1.16637 x 10-5 GeV-2

This structure known since Fermi

Now include Leptons

Simplest case, include an SU(2) doublet of left-handed leptons

Right-handed electron, eR=(1+5)e/2, is SU(2) singlet No right-handed neutrino

eeL

L

L

)1(2

1

)1(2

1

5

5

*Standard Model has massless neutrinos—discovery of non-zero neutrino mass evidence for physics beyond the SM

Leptons, 2

Couple gauge fields to leptons

LiiL

RRleptons

Wg

iYBg

ii

eYBg

iieL

22

2

Leptons, 3

Write in terms of charged and neutral currents

emZleptons JeAJZJWJWgkineticL )(

RWRLWLLLW

Z

LL

emem

eeeeJ

eJ

eeQJ

22 sin2sin21cos2

12

1

Muon decay

Consider e e

Fermi Theory:

e

e

• EW Theory:

eF uuuugGie

2

1

2

122 55

eW

uuuugMk

ige

2

1

2

11

255

22

2

For k<< MW, 22GF=g2/2MW2

For k>> MW, 1/E2

e

e

W

22

2

2

1

82 vM

gG

W

F

Parameters of SU(2) x U(1) Sector

g, g',, Trade for: =1/137.03599911(46) from (g-2)e and

quantum Hall effect GF=1.16637(1) x 10-5 GeV-2 from muon lifetime MZ=91.18750.0021 GeV Plus Higgs and fermion masses

Now Add Quarks to Standard Model Include color triplet quark doublet

Right handed quarks are SU(2) singlets, uR=(1+5)u, dR=(1+5)d

With weak hypercharge YuR=4/3, YdR=-2/3, YQL=1/3

iL

iLL d

uQ

Qem=(I3+Y)/2

i=1,2,3 for color

Quarks, 2

Couplings of charged current to W and Z’s take the form:

uWddWu

gLWqq )1()1(

2255

qZRLq

gL qq

WZqq )1()1(

cos4 55

Wemq

Wemq

QR

QIL

2

23

sin2

sin2

What about fermion masses?

Fermion mass term:

Left-handed fermions are SU(2) doublets

Scalar couplings to fermions:

Effective Higgs-fermion coupling

Mass term for down quark:

LRRLmmL

..chdQL RLdd

L

L d

uQ

..0

),(2

1chd

hvduL RLLdd

v

M dd

2

Forbidden by SU(2)xU(1) gauge invariance

Fermion Masses, 2

Mu from c=i2* (not allowed in SUSY)

For 3 generations, , =1,2,3 (flavor indices)

0

c

hcuQL RcLu v

M uu

2

,

..2

)(chdduu

hvL RLdRLuY

Fermion masses, 3

Unitary matrices diagonalize mass matrices

Yukawa couplings are diagonal in mass basis Neutral currents remain flavor diagonal Not necessarily true in models with extended Higgs

sectors

mRdR

mRuR

mLdL

mLuL

dVduVu

dUduUu

• Charged current:m

LdumLLL dVUuduJ

)(

2

1

2

1

CKM matrix

Basics Four free parameters in gauge-Higgs sector

Conventionally chosen to be =1/137.0359895(61) GF =1.16637(1) x 10-5 GeV -2

MZ=91.1875 0.0021 GeV MH

Express everything else in terms of these parameters

22

22

2

1282

WZ

WW

F

MM

MM

gG

Predicts MW

Inadequacy of Tree Level Calculations Mixing angle is predicted quantity

On-shell definition cos2W=MW2/MZ

2

Predict MW

Plug in numbers: MW predicted =80.939 GeV

MW(exp) =80.404 0.030 GeV

Need to calculate beyond tree level

2

22

ZFWW MG

cs

1

2

2

2

4112

ZFFW

MGGM

Modification of tree level relations

rMG

WW

F

1

1

sin2 22

W

WtFt mGr

2

2

2

2

sin

cos

28

3

Contributions to r from top quark and Higgs loops

6

5ln

224

112

2

2

2

W

hWFh

M

MMGr

r is a physical quantity which incorporates 1-loop corrections

Extreme sensitivity of precision measurements to mt

Where are we with Z’s?

At the Z pole: 2 x 107 unpolarized Z’s at LEP 5 x 105 Z’s at SLD with Pe 75%

What did we measure at the Z? Z lineshape , Z, MZ

Z branching ratios Asymmetries

W+W- production at 200 GeV Searches for Zh

e+e-ff

)1(2

222

zs

QNffee

dz

d fc

zLRLR

zLRLR

MMs

sMGN

ffee

ffee

ZZZ

ZFc

))((2

)1)()((

)(64 2222

22222

2222

42

zLRLRzLRLRMMs

MsMQNffeeffee

ZZZ

ZZfc

2)1(

)(

)( 22222

22

exchange

-Z interference

Changes sign at pole

Z exchangez=cos

e+e-ff (#2)

Assume energy near the Z-pole, so include only Z exchange

zLRLRzLRLRMMs

sMGNffee

dz

dffeeffee

ZZZ

ZFc ))((2)1)()(()(64

2222222222222

42

Contributes only to asymmetries if acceptance is symmetric

))((24

22222

42

ffeeZ

ZFcpeakZ LRLR

MGNffee

Z cross section

Number of light neutrinos: N=2.98400.0082

Z

MZ

Requires precise calibration of energy of machine

TevatronTevatron running pp at s=2 TeV

Scheduled to shut down 2009-2010

Z’s at the Tevatron

Z-production Amplitude has pole at MZ

Invariant mass distribution of e+e-

eeZqq

222Zeeee Mppm

2)(

1

ee pp

W’s at the Tevatron

Consider We Invariant mass of the leptonic system

Missing transverse energy of neutrino inferred from observed momenta

Can’t reconstruct invariant mass Define transverse mass observable

2222 )( zezTeTee ppppEEm

)cos1(22

222

TeTTeT

TeTTeTT

EEpp

ppEEm

Statistics enough to best LEP 2

W Mass Measurement

Location of peak gives MW

Shape of distribution sensitive to W

World Average for W mass

Direct measurements (Tevatron/LEP2) and indirect measurements (LEP1/SLD) in excellent agreement

Indirect measurements assume a Higgs mass

LEPEWWG home page, 2006

Data prefer light Higgs

Low Q2 data not included Doesn’t include atomic parity

violation in cesium, parity violation in Moller scattering, & neutrino-nucleon scattering (NuTeV)

Higgs fit not sensitive to low Q2

data Mh< 207 GeV

1-side 95% c.l. upper limit, including direct search limit

(Mh < 166 GeV ICHEP 2006) Direct search limit from e+e-Zh

Top Quark mass pins down Higgs Mass

Data prefer a light Higgs

2006

Understanding Higgs Limit

1118.0

)(085.0

1172

525.0102761.0

)(5098.0

100ln008.0

100ln0579.0364.80

2)5(

2

Zs

tZhad

hhW

M

GeV

MM

GeV

M

GeV

MM

MW(experiment)=80.404 0.030

sin2w depends on scale

Moller scattering, e-e-e-e-

-nucleon scattering Atomic parity violation in

Cesium

We have a model….And it works to the 1% level

Gives us confidence to predict the future!

Electroweak Theory is Precision Theory

2006